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Comment by catnaroek | Hacker News Reader

> You should read Russell on definite descriptions

An expression is only meaningful in a context in which its free variables are bound. So, in a context in which the concept of “prime number” doesn't exist, any statement about “prime numbers” is meaningless.

> As for your distinction between philosophical truth, and mathematical truth, you should elaborate

Philosophical truth is the correspondence between a statement and reality. Mathematical truth, at least to a formalist like me, is the syntactic property of being substitutable with the symbol “true” in a given context.

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Philosophers of math aren't confused by this. What they are asking is whether "mathematical truth" has any connection with truth. Formalists more or less deny this, Platonists affirm it.

As a side note, correspondence with reality is a contentious definition of truth. It's just one theory of what truth is.

I don't really know what to say about your claims about meaninglessness, except that I think they're false. But I don't even know where to start in arguing that. And while I don't think this is much of an argument, I don't think there are really any philosophers, mathematicians, or logicians who have worked out any theory along those lines.

> I don't really know what to say about your claims about meaninglessness, except that I think they're false.

Syntax has no meaning outside of the rules for manipulating it. If no rule applies to a particular expression, then it has no meaning whatsoever. And, in any sensible formal system, like typed lambda calculi with Church-style semantics, expressions only have meaning in contexts where their free variables are bound.

It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx.

The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it.

This is much like the mistake that Russell criticized in On Denoting, with the difference that he discussed definite descriptions. But the error is the same: he attacked opponents who thought that "the present king of france is bald" was either meaningless, or required some sort of bizzaro existence for the present King of France. And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.

> It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx.

This is a very low insult - to assume I don't understand the distinction between “F(a)”, where “a” appears free, and “Ex. F(x)”, where “x” appears bound. I understand logical quantifiers and variable binding just fine, thanks.

> The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it.

Another very low insult - to assume that I would conflate “false” with “meaningless”. “There is a natural number that is both even and odd” is false. “There is a myppit inside the quxxit” is meaningless unless we first define: (0) what a “myppit” is, (1) what a “quxxit” is, (2) what specific “quxxit” we're talking about.

The one who's making a mistake is you, because “F” isn't the same as “F(x)”, but rather “λx.F(x)”. In a context where the variable “x” isn't bound, the the latter is meaningful, but the former is not. In other words, the judgment “Γ ⊢ F(x) : Prop”, asserting that “F(x)” is indeed a proposition in the context “Γ”, shouldn't be derivable if “Γ” doesn't contain “x”. But the judgment “Γ ⊢ λx.F(x) : T → Prop”, asserting that “F” is a predicate ranging over “T”s, is derivable from “Γ, x:T ⊢ F(x) : Prop”, and “Γ, x:T” is a context containing “x”.

> And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.

Unfortunately, “Bald(king)“ isn't the same as “E!person. King(person) ∧ Bald(person)”.

Regarding insults, you started by suggesting that philosophers of math were ignorant of the distinction between truth as either proof theoretic derivation from axioms, or truth in a model and truth simpliciter, I don't think you have room to worry about who is being insulting. Philosophers typically would be introduced to these concepts in advanced undergraduate logic or early on in grad school.

As for the rest of it, I'll leave with one final thought: as a meta-philosophical point, we should rarely attribute meaninglessness to statements unless we are forced to. And it's easy enough to show how to interpret existentials as meaningful even in the absence of a referent. You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.

> we should rarely attribute meaninglessness to statements unless we are forced to.

You don't “attribute meaninglessness” to statements. Syntax is a priori meaningless, and the rules for manipulating it give it meaning. “Meaninglessness” is thus just the lack of a given meaning.

> You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.

Here's a good reason: leveraging the power and reliability of computers for doing mathematics, which is only possible if syntax is mechanically interpretable. The kind of word tricks you suggest, like interpreting a sentence containing the phrase “the King of France” as implicitly asserting the existence (and uniqueness) of a King of France (in addition to whatever you explicitly say about him), are incompatible with a nice, compositional, crystal clear, reliable, mechanical interpretation of syntax.

The concept of prime numbers and the prime numbers themselves is a different thing.

Unicorns don't exist but statements about them aren't meaningless. Fictionalism accepts the concept of mathematical entities but denies their existence in the real world.

> The concept of prime numbers and the prime numbers themselves is a different thing.

I'm okay with that, and haven't said anything suggesting otherwise. However, if you want to reject the existence of mathematical objects in the real world, you have to reject the existence of mathematical definitions too, because mathematical definitions are mathematical objects just like any other. So, if the definition of “prime number” doesn't exist in the real world, then any statement about prime numbers is meaningless in the real world.

> Unicorns don't exist but statements about them aren't meaningless.

Unicorns don't exist in the real world, but the definition of “unicorn” does - just grab any dictionary. It's on the basis the existing definition that you can say “unicorns don't exist”. On the other hand, you may reject the definition of “horse”, but, if you don't, you have to accept that horses exist in the real world. And, if you do reject it, then you also have to reject statements about horses as completely meaningless.

> Fictionalism accepts the concept of mathematical entities but denies their existence in the real world.

The question of whether mathematical objects exist in the real world is ill-posed in the first place. It's like asking if love can be stored in boxes.

At some point mathematical objects are defined with meta-mathematical definitions (axioms etc), so we can accept definitions based on those terms.

But in any case. You obviously do not believe that mathematical objects exist in the first place and it the question is ill posed based on your concept of reality and existence.

That's understandable, but the debate is about which is the right definition of existence reality etc. If you are not interested in this discussion and know the answers then you have nothing to gain from this paper or this discussion.

> At some point mathematical objects are defined with meta-mathematical definitions (axioms etc), so we can accept definitions based on those terms.

“Metamathematics” doesn't exist in isolation from the rest of mathematics. If you reject the existence of mathematical objects (in the real world or elsewhere), you must also reject the existence of metamathematical definitions (in the same context), because the former include the latter.

> But in any case. You obviously do not believe that mathematical objects exist in the first place and it the question is ill posed based on your concept of reality and existence.

I only denied the meaningfulness of asking whether mathematical objects exist in the real world. The terms “real world”, “reality”, etc., I prefer to reserve for that which can be apprehended through physical experience. The distinguishing feature of reality is that you can't reject its existence (in the same way I hypothesized rejecting the definition of the word “horse” in my previous comment) - your senses force you to accept it. If someone cuts through your skin with the intention to bleed you to death, you will feel pain. You may resign yourself to your fate, but that won't make the pain go away. On the other hand, if you find it useless, annoying, etc. to do mathematics, all you need to do is stop doing it.

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